What is ##7 \times 7## in base ##7##?

[RChristenk]

Homework Statement

What is $7 \times 7$ in base $7$?

Relevant Equations

Change radix to $7$

In base $7$, only the digits $0,1,2,3,4,5,6$ can be used. So $7 \times 7 = 49$ in base $10$, but $49$ is a multiple of $7$ seven times, so at the right-most digit position of the answer, I can put down a $0$ and carry a $7$ to the next-left digit position. I can't put down $7$ in this digit position because it is bigger than the largest allowable digit, $6$, so I put down a $0$ again, giving me a partial answer of _$00$ at this step. Seven goes into seven $1$ time, so now I can put down $1$ in the next-left position and complete this question. Hence the answer is $100$, or that in base $7$, $7 \times 7 = 100$.

My first question is how can $7 \times 7$ be a valid question if everything is to be in base $7$ (only zero to six can be used)? Or is this actually asking $7 \times 7$ in base $10$ and for the answer to be converted into base $7$? I'm very confused because in the intermediary calculations I'm using the digit seven but calculating according to base $7$ rules.

My second question is how do I write down base $7$ multiplication in long hand on paper? Would something like this be correct:

$\begin{array}{c} 1&7& \\ \hline & &7 \\ & &7 \\ \hline 1& 0&0 \end{array}$

I'm mostly confused here by how to treat the initial $49$, but I couldn't see a logical place to write it down so I just went ahead and assumed seven goes into forty-nine seven times and wrote down $7$ at the very top.

 

[Orodruin]

My first question is how can 7×7 be a valid question if everything is to be in base 7 (only zero to six can be used)?

It isn’t.

Or is this actually asking 7×7 in base 10 and for the answer to be converted into base 7?

We cannot know this. Only you have access to the source of the question.

 

[fresh_42]

Homework Statement: What is $7 \times 7$ in base $7$?
Relevant Equations: Change radix to $7$

In base $7$, only the digits $0,1,2,3,4,5,6$ can be used. So $7 \times 7 = 49$ in base $10$, but $49$ is a multiple of $7$ seven times, so at the right-most digit position of the answer, I can put down a $0$ and carry a $7$ to the next-left digit position. I can't put down $7$ in this digit position because it is bigger than the largest allowable digit, $6$, so I put down a $0$ again, giving me a partial answer of _$00$ at this step. Seven goes into seven $1$ time, so now I can put down $1$ in the next-left position and complete this question. Hence the answer is $100$, or that in base $7$, $7 \times 7 = 100$.

My first question is how can $7 \times 7$ be a valid question if everything is to be in base $7$ (only zero to six can be used)? Or is this actually asking $7 \times 7$ in base $10$ and for the answer to be converted into base $7$? I'm very confused because in the intermediary calculations I'm using the digit seven but calculating according to base $7$ rules.

My second question is how do I write down base $7$ multiplication in long hand on paper? Would something like this be correct:

$\begin{array}{c} 1&7& \\ \hline & &7 \\ & &7 \\ \hline 1& 0&0 \end{array}$

I'm mostly confused here by how to treat the initial $49$, but I couldn't see a logical place to write it down so I just went ahead and assumed seven goes into forty-nine seven times and wrote down $7$ at the very top.

Such a representation of a number $a$ to a base $b$ is defined as
$$
a=a_n\cdot b_n^n + \ldots + a_2\cdot 7^2+a_1\cdot 7^1+a_0\cdot 7^0
$$
with $a_n,\ldots ,a_2,a_1,a_0 \in \{0,1,2,\ldots, b-1\}.$

This means that seven in base seven is written $10=1\cdot 7 + 0\cdot 1.$ Therefore seven times seven is
$\begin{array}{c} 1\,0\,\;& \cdot & 1\,0\,\; \\ \hline 1\,0\,\;& & \\ \,0\,0\,0 & &\\ \hline \,1\,0\,0&& \end{array}$
and $100=1\cdot 7^2 +0\cdot 7^1=0\cdot 7^0=1\cdot 49+0\cdot 7 +0\cdot 1.$

The multiplication is the same, only with fewer numerals.

 

[DaveC426913]

My first question is how can $7 \times 7$ be a valid question if everything is to be in base $7$

The question is asking what is 7₁₀x7₁₀ in base 7?
Because if it were asking it in base 7 then the question would be what is 10₇ x 10₇? ("10" being 7 - in base 7)
And the answer would be 100₇ - which is 49₁₀.

 

[Mark44]

What is $7 \times 7$ in base 7?

It's reasonable to assume that the 7s above are decimal numbers (i.e., in base-10), so I agree with others that the answer is $100_7$.

 

[Hill]

Perhaps I am too literal, but to me "$7 \times 7$ in base $7$" is $10 \times 10$.

 

[fresh_42]

Perhaps I am too literal, but to me "$7 \times 7$ in base $7$" is $10 \times 10$.

See my post #3.

 

[SammyS]

It's reasonable to assume that the 7s above are decimal numbers (i.e., in base-10), so I agree with others that the answer is $100_7$.

Me too.

Perhaps a better wording of the problem is to use only words.

What is seven times seven ? Write the result in base seven.

 

[DaveC426913]

Me too.

Perhaps a better wording of the problem is to use only words.

What is seven times seven ? Write the result in base seven.

7_n x 7_n = 49₁₀
for any n larger than 7.

And we know n can't be 7 or less, since that is invalid.

Therefore, 7x7 is actually unambiguous.

7₅ x 7₅ = n/a
7₇ x 7₇ = n/a
7₉ x 7₉ = 49₁₀
7₁₇ x 7₁₇ = 49₁₀

In fact:

7₂₃ x 7₄₇ = 49 ₁₀

 

[fresh_42]

One could write the numbers as $[a]_b$ but it is pretty obvious from the context what is meant. If we are talking about $7$ in the base of $7$ then it is clear that $[7]_{10}=[10]_7$ is meant and that $10\cdot 10 =49$ is short for $[10]_7\cdot [10]_7=[100]_7=[49]_{10}.$

There's no need to search through the crumbs.

 

[gmax137]

I think "b times b, in base b" is always "100" that's how I solved it, anyway.

 

[WWGD]

One could write the numbers as $[a]_b$ but it is pretty obvious from the context what is meant. If we are talking about $7$ in the base of $7$ then it is clear that $[7]_{10}=[10]_7$ is meant and that $10\cdot 10 =49$ is short for $[10]_7\cdot [10]_7=[100]_7=[49]_{10}.$

There's no need to search through the crumbs.

Still, if we work with classes, rather than numbers, which representative do we choose? [10]_7=[17]_7=...
Maybe I'm nitpicking?

 

[fresh_42]

Still, if we work with classes, rather than numbers, which representative do we choose? [10]_7=[17]_7=...
Maybe I'm nitpicking?

These weren't meant to be classes, only a notation that allows one to interpret the digits of $a$ according to different bases $b$. We could write $10_7=7_{10}$ but I thought $[10]_7=[7]_{10}$ is easier to read. Rectangular brackets are not automatically equivalence classes.

 

[WWGD]

These weren't meant to be classes, only a notation that allows one to interpret the digits of $a$ according to different bases $b$. We could write $10_7=7_{10}$ but I thought $[10]_7=[7]_{10}$ is easier to read. Rectangular brackets are not automatically equivalence classes.

Guess a classic/clear example of notation overload. Notation can only do, help, so much.